OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Elasticity.global_uniqueness, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, result family 372, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass of lean/OAI/MathematicalPhysics/Elasticity/BoundaryData.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryPacket, with open Set MeasureTheory Filter, open scoped Topology BigOperators, open Elasticity ElasticityBoundaryPhysicalTransfer ElasticityBoundaryForceSupport):

lemma Setup.field_gradient_mass (s : Setup Ω x) (hΩ : MeasurableSet Ω) (q : Polynomial W) :
    ∃ C : ℝ, 0≤C ∧ ∀ δ : ℝ, 0<δ → δ≤1 → ∀ i d,
      (∫ z in Ω, ‖dd (s.field q δ i) (basis d) z‖^2)≤C

Definitions. The statement is written with OAI.ElasticityBoundaryChainRule.dd (definitions bundle OAIQ6El000 of this split); OAI.ElasticityBoundaryChartOperator.I, OAI.ElasticityBoundaryChartOperator.basis, OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryPacket.I, OAI.ElasticityBoundaryPacket.Setup, OAI.ElasticityBoundaryPacket.X and OAI.ElasticityBoundaryPacket.Setup.field (definitions bundle OAIQ6El001 of this split).

Used by. The proof of OAI.ElasticityBoundaryPacket.Setup.trial_size of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El000
import Definitions.Def_OAIQ6El001

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

namespace ElasticityBoundaryPacket

open Set MeasureTheory Filter
open scoped Topology BigOperators
open Elasticity ElasticityBoundaryPhysicalTransfer ElasticityBoundaryForceSupport
  ElasticityBoundaryCutoffLayer ElasticityBoundaryCutoffParametrix ElasticityBoundaryPacketMass
  ElasticityBoundaryGradientMass ElasticityBoundaryGradientTransfer ElasticityBoundaryPhysicalMass
  ElasticityBoundaryChainRule ElasticityBoundaryChartOperator ElasticityBoundaryStrongLayer
  ElasticityBoundaryLocalParametrix ElasticityBoundaryMVNormal ElasticityBoundaryActualTower
  ElasticityBoundaryProfileBounds ElasticityBoundaryLayerOperator

variable {Ω : Set X} {x : X}

lemma Setup.field_gradient_mass (s : Setup Ω x) (hΩ : MeasurableSet Ω) (q : Polynomial W) :
    ∃ C : ℝ, 0≤C ∧ ∀ δ : ℝ, 0<δ → δ≤1 → ∀ i d,
      (∫ z in Ω, ‖dd (s.field q δ i) (basis d) z‖^2)≤C := by
  sorry

end ElasticityBoundaryPacket

end

end OAI
end

Source

OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 372; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Elasticity/BoundaryData.lean (declaration OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass)

View graph